Determining whether a quadratic has a maximum or minimum

Question

So I've learnt that quadratics equations with a positive coefficient on the squared term have a minimum and a maximum if the coefficient is negative. But if we rearrange the quadratic and change the signs of the squared term, doesn't that mean the equation's maximum would change to a minimum and vice versa? e.g $ x^2 + 5x + 10 = 0$ becomes $-x^2 - 5x - 10 =0 $ even though we've not changed the equation, it's now got a maximum rather than a minimum. What am I misunderstanding here?

Answer 1

If $f(x)=0$, the maximum and minimum of $f(x)$ are both $0$ (unless there are no $x$ such that $f(x)=0$).